arXiv subjects

John L. Friedman

Publications and source records attributed to John L. Friedman.

At least 19 recordsLinked to original sources

arXiv2026

Notes on Gravitational Physics

These notes are self-contained, with the first 7 chapters used in a one-semester course with recommended texts by Wald, by Misner, Thorne and Wheeler, and by Schutz. In its treatment of topics covered in these standard texts, the presentation here typically includes steps skipped in Wald or MTW. Treatments of gravitational waves, particle orbits in black-hole backgrounds, the Teukolsky equation, and the initial value equations are motivated in part by the discoveries of gravitational waves from the inspiral and coalescence of binary black holes and neutron stars, advances in numerical relativity, and the expected LISA space-based observatory. The notes begin with a detailed presentation of special relativity with a geometrical orientation, starting with with time dilation and length contraction and including relativistic particles, fluids, electromagnetism, and curvilinear coordinates. Chaps. 2-5 cover curvature, the Einstein equation, relativistic stars, and black holes. Chap. 6, on gravitational waves, includes a discussion of detection and noise. Chap. 7 is a brief introduction to cosmology, deriving the metrics of homogeneous isotropic space, the equations governing a universe with matter, radiation and vacuum energy, and their solutions, and discussions of the cosmological redshift and on using gravitational waves to measure the Hubble constant. Chap. 8, on the initial value problem, has a section on the form of the equations used in numerical relativity. The Newman-Penrose formalism and the Teukolsky equation are covered in Chap. 9. Following that is a chapter on black-hole thermodynamics and a final chapter on the gravitational action and on conserved quantities for asymptotically flat spacetimes, using Noether's theorem. An appendix covers forms, densities, integration, and Cartan calculus.

gr-qc↗

arXiv2022

Astrophysical Implications of Neutron Star Inspiral and Coalescence

The first inspiral of two neutron stars observed in gravitational waves was remarkably close, allowing the kind of simultaneous gravitational wave and electromagnetic observation that had not been expected for several years. Their merger, followed by a gamma-ray burst and a kilonova, was observed across the spectral bands of electromagnetic telescopes. These GW and electromagnetic observations have led to dramatic advances in understanding short gamma-ray bursts; determining the origin of the heaviest elements; and determining the maximum mass of neutron stars. From the imprint of tides on the gravitational waveforms and from observations of X-ray binaries, one can extract the radius and deformability of inspiraling neutron stars. Together, the radius, maximum mass, and causality constrain the neutron-star equation of state, and future constraints can come from observations of post-merger oscillations. We selectively review these results, filling in some of the physics with derivations and estimates.

astro-ph.HE↗

arXiv2018

Differential rotation of the unstable nonlinear r-modes

At second order in perturbation theory, the $r$-modes of uniformly rotating stars include an axisymmetric part that can be identified with differential rotation of the background star. If one does not include radiation-reaction, the differential rotation is constant in time and has been computed by Sá. It has a gauge dependence associated with the family of time-independent perturbations that add differential rotation to the unperturbed equilibrium star: For stars with a barotropic equation of state, one can add to the time-independent second-order solution arbitrary differential rotation that is stratified on cylinders (that is a function of distance $\varpi$ to the axis of rotation). We show here that the gravitational radiation-reaction force that drives the $r$-mode instability removes this gauge freedom: The expontially growing differential rotation of the unstable second-order $r$-mode is unique. We derive a general expression for this rotation law for Newtonian models and evaluate it explicitly for slowly rotating models with polytropic equations of state.

astro-ph.HE↗

arXiv2018

Limits on Magnetic Field Amplification from the r-Mode Instability

At second order in perturbation theory, the unstable r-mode of a rotating star includes growing differential rotation whose form and growth rate are determined by gravitational-radiation reaction. With no magnetic field, the angular velocity of a fluid element grows exponentially until the mode reaches its nonlinear saturation amplitude and remains nonzero after saturation. With a background magnetic field, the differential rotation winds up and amplifies the field, and previous work where large mode amplitudes were considered suggests that the amplification may damp out the instability. A background magnetic field, however, turns the saturated time-independent perturbations corresponding to adding differential rotation into perturbations whose characteristic frequencies are of order the Alfvén frequency. As found in previous studies, we argue that magnetic- field growth is sharply limited by the saturation amplitude of an unstable mode. In contrast to previous work, however, we show that if the amplitude is small, i.e., of order 10^(-4), then the limit on the magnetic-field growth is stringent enough to prevent the loss of energy to the magnetic field from damping or significantly altering an unstable r-mode in nascent neutron stars with normal interiors and in cold stars whose interiors are type II superconductors. We show this result first for a toy model, and we then obtain an analogous upper limit on magnetic field growth using a more realistic model of a rotating neutron star. Our analysis depends on the assumption that there are no marginally unstable perturbations, and this may not hold when differential rotation leads to a magnetorotational instability.

astro-ph.HE↗

arXiv2017

Magnetic field amplification by the r-mode instability

We discuss magnetic field enhancement by unstable r-modes (driven by the gravitational radiation reaction force) in rotating stars. In the absence of a magnetic field, gravitational radiation exponentially increases the r-mode amplitude $α$, and accelerates differential rotation (secular motion of fluid elements). For a magnetized star, differential rotation enhances the magnetic field energy. Rezzolla et al. (2000--2001) argued that if the magnetic energy grows faster than the gravitational radiation reaction force pumps energy into the r-modes, then the r-mode instability is suppressed. Chugunov (2015) demonstrated that without gravitational radiation, differential rotation can be treated as a degree of freedom decoupled from the r-modes and controlled by the back reaction of the magnetic field. In particular, the magnetic field windup does not damp r-modes. Here we discuss the effect of the back reaction of the magnetic field on differential rotation of unstable r-modes, and show that it limits the generated magnetic field and the magnetic energy growth rate preventing suppression of the r-mode instability by magnetic windup at low saturation amplitudes, $α\ll 1$, predicted by current models.

astro-ph.HE↗

arXiv2017

Upper limit set by causality on the tidal deformability of a neutron star

A principal goal of gravitational-wave astronomy is to constrain the neutron star equation of state (EOS) by measuring the tidal deformability of neutron stars. The tidally induced departure of the waveform from that of point-particle (or spinless binary black hole (BBH)) increases with the stiffness of the EOS. We show that causality (the requirement that the speed of sound is less than the speed of light for a perfect fluid satisfying a one-parameter equation of state) places an upper bound on tidal deformability as a function of mass. Like the upper mass limit, the limit on deformabity is obtained by using an EOS with $v_{sound} = c$ for high densities and matching to a low density (candidate) EOS at a matching density of order nuclear saturation density. We use these results and those of [B.D. Lackey et al., Phys. Rev. D 89, 043009 (2014)] to estimate the resulting upper limit on the gravitational-wave phase shift of a black hole-neutron star (BHNS) binary relative to a BBH. Even for assumptions weak enough to allow an maximum mass of $4\ M_\odot$ (a match at nuclear saturation density to an unusually stiff low-density candidate EOS), the upper limit on dimensionless tidal deformability is stringent. It leads to a still more stringent estimated upper limit on the maximum tidally induced phase shift prior to merger. We comment in an appendix on the relation between causality, the condition $v_{sound} < c$, and the condition $dp/dε< 1$ for the effective EOS governing the equilibrium star.

gr-qc↗

arXiv2014

Gravitational Self-force in a Radiation Gauge

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin-weight $\pm 2$ perturbed Weyl scalar ($ψ_0$ or $ψ_4$) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from $ψ_0$ in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate $r_0$ of the particle in any convenient gauge. The resulting metric perturbation is singular on the trajectory of the particle and discontinuous across the sphere $r=r_0$. A mode-sum method can be used to renormalize the self-force, but the justification given in the published version of this paper \cite{sf2} referred to work by Sam Gralla \cite{gralla10} to justify the use of the renormalized self-force, and the radiation gauge we use does not satisfy the regularity conditions required by Gralla. Instead we show that the renormalized self-force, computed either from the retarded field for $r>r_0$ or for $r r_0$ and for $r<r_0$ gives the correct equation of motion for our gauge (what Pound et al. call the no-string gauge).

gr-qc↗

arXiv2014

Combined gravitational and electromagnetic self-force on charged particles in electrovac spacetimes

We consider the self-force on a charged particle moving in a curved spacetime with a background electromagnetic field, extending previous studies to situations in which gravitational and electromagnetic perturbations are comparable. The formal expression $f^{ret}_α$ for the self-force on a particle, written in terms of the retarded perturbed fields, is divergent, and a renormalization is needed to find the particle's acceleration at linear order in its mass $m$ and charge $e$. We assume that, as in previous work in a Lorenz gauge, the renormalization for accelerated motion comprises an angle average and mass renormalization. Using the short distance expansion of the perturbed electromagnetic and gravitational fields, we show that the renormalization is equivalent to that obtained from a mode sum regularization in which one subtracts from the expression for the self-force in terms of the retarded fields a singular part field comprising only the leading and subleading terms in the mode sum. The most striking part of our result, arising from a remarkable cancellation, is that the renormalization involves no mixing of electromagnetic and gravitational fields. In particular, the renormalized mass is obtained by subtracting (1) the purely electromagnetic contribution from a point charge moving along an accelerated trajectory and (2) the purely gravitational contribution from a point mass moving along the same trajectory. In a mode-sum regularization, the same cancellation implies that the required regularization parameters are sums of their purely electromagnetic and gravitational values.

gr-qc↗

arXiv2014

Self force on an accelerated particle

We calculate the singular field of an accelerated point particle (scalar charge, electric charge or small gravitating mass) moving on an accelerated (non-geodesic) trajectory in a generic background spacetime. Using a mode-sum regularization scheme, we obtain explicit expressions for the self-force regularization parameters. In the electromagnetic and gravitational case, we use a Lorenz gauge. This work extends the work of Barack and Ori [1] who demonstrated that the regularization parameters for a point particle in geodesic motion in a Schwarzschild spacetime can be described solely by the leading and subleading terms in the mode-sum (commonly known as the $A$ and $B$ terms) and that all terms of higher order in $\ell$ vanish upon summation (later they showed the same behavior for geodesic motion in Kerr [2], [3]). We demonstrate that these properties are universal to point particles moving through any smooth spacetime along arbitrary (accelerated) trajectories. Our renormalization scheme is based on, but not identical to, the Quinn-Wald axioms. As we develop our approach, we review and extend work showing that that different definitions of the singular field used in the literature are equivalent to our approach. Because our approach does not assume geodesic motion of the perturbing particle, we are able use our mode-sum formalism to explicitly recover a well-known result: The self-force on static scalar charges near a Schwarzschild black hole vanishes.

gr-qc↗

arXiv2013

Matter effects on binary neutron star waveforms

Using an extended set of equations of state and a multiple-group multiple-code collaborative effort to generate waveforms, we improve numerical-relativity-based data-analysis estimates of the measurability of matter effects in neutron-star binaries. We vary two parameters of a parameterized piecewise-polytropic equation of state (EOS) to analyze the measurability of EOS properties, via a parameter Λ that characterizes the quadrupole deformability of an isolated neutron star. We find that, to within the accuracy of the simulations, the departure of the waveform from point-particle (or spinless double black-hole binary) inspiral increases monotonically with Λ, and changes in the EOS that did not change Λ are not measurable. We estimate with two methods the minimal and expected measurability of Λ in second- and third- generation gravitational-wave detectors. The first estimate, using numerical waveforms alone, shows two EOS which vary in radius by 1.3km are distinguishable in mergers at 100Mpc. The second estimate relies on the construction of hybrid waveforms by matching to post-Newtonian inspiral, and estimates that the same EOS are distinguishable in mergers at 300Mpc. We calculate systematic errors arising from numerical uncertainties and hybrid construction, and we estimate the frequency at which such effects would interfere with template-based searches.

gr-qc↗

arXiv2013

Extracting equation of state parameters from black hole-neutron star mergers: aligned-spin black holes and a preliminary waveform model

Information about the neutron-star equation of state is encoded in the waveform of a black hole-neutron star system through tidal interactions and the possible tidal disruption of the neutron star. During the inspiral this information depends on the tidal deformability Lambda of the neutron star, and we find that Lambda is the best measured parameter during the merger and ringdown as well. We performed 134 simulations where we systematically varied the equation of state as well as the mass ratio, neutron star mass, and aligned spin of the black hole. Using these simulations we have developed an analytic representation of the full inspiral-merger-ringdown waveform calibrated to these numerical waveforms, and we use this analytic waveform to estimate the accuracy to which Lambda can be measured with gravitational-wave detectors. We find that although the inspiral tidal signal is small, coherently combining this signal with the merger-ringdown matter effect improves the measurability of Lambda by a factor of ~3 over using just the merger-ringdown matter effect alone. However, incorporating correlations between all the waveform parameters then decreases the measurability of Lambda by a factor of ~3. The uncertainty in Lambda increases with the mass ratio, but decreases as the black hole spin increases. Overall, a single Advanced LIGO detector can measure Lambda for mass ratios Q = 2--5, black hole spins J_BH/M_BH^2 = -0.5--0.75, neutron star masses M_NS = 1.2M_sun--1.45M_sun, and an optimally oriented distance of 100Mpc to an uncertainty of ~10%--100%. For the proposed Einstein Telescope, the uncertainty in Lambda is an order of magnitude smaller.

gr-qc↗

arXiv2012

EMRI corrections to the angular velocity and redshift factor of a mass in circular orbit about a Kerr black hole

This is the first of two papers on computing the self-force in a radiation gauge for a particle moving in circular, equatorial orbit about a Kerr black hole. In the EMRI (extreme-mass-ratio inspiral) framework, with mode-sum renormalization, we compute the renormalized value of the quantity $h_{αβ}u^αu^β$, gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; and we find the related order $\frak{m}$ correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity). The radiative part of the perturbed metric is constructed from the Hertz potential which is extracted from the Weyl scalar by an algebraic inversion\cite{sf2}. We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics and use mode-sum renormalization to find the renormalization coefficients by matching a series in $L=\ell+1/2$ to the large-$L$ behavior of the expression for $H := \frac12 h_{αβ}u^αu^β$. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge.

gr-qc↗

arXiv2011

Neutron star equation of state via gravitational wave observations

Gravitational wave observations can potentially measure properties of neutron star equations of state by measuring departures from the point-particle limit of the gravitational waveform produced in the late inspiral of a neutron star binary. Numerical simulations of inspiraling neutron star binaries computed for equations of state with varying stiffness are compared. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly if the neutron stars are more compact. This suggests that gravitational wave observations at frequencies around 1 kHz will be able to measure a compactness parameter and place stringent bounds on possible neutron star equations of state. Advanced laser interferometric gravitational wave observatories will be able to tune their frequency band to optimize sensitivity in the required frequency range to make sensitive measures of the late-inspiral phase of the coalescence.

gr-qc↗

arXiv2011

Extracting equation of state parameters from black hole-neutron star mergers. I. Nonspinning black holes

The late inspiral, merger, and ringdown of a black hole-neutron star (BHNS) system can provide information about the neutron-star equation of state (EOS). Candidate EOSs can be approximated by a parametrized piecewise-polytropic EOS above nuclear density, matched to a fixed low-density EOS; and we report results from a large set of BHNS inspiral simulations that systematically vary two parameters. To within the accuracy of the simulations, we find that, apart from the neutron-star mass, a single physical parameter Lambda, describing its deformability, can be extracted from the late inspiral, merger, and ringdown waveform. This parameter is related to the radius, mass, and l=2 Love number, k_2, of the neutron star by Lambda = 2k_2 R^5/3M_{NS}^5, and it is the same parameter that determines the departure from point-particle dynamics during the early inspiral. Observations of gravitational waves from BHNS inspiral thus restrict the EOS to a surface of constant Lambda in the parameter space, thickened by the measurement error. Using various configurations of a single Advanced LIGO detector, we find that neutron stars are distinguishable from black holes of the same mass and that Lambda^{1/5} or equivalently R can be extracted to 10-40% accuracy from single events for mass ratios of Q=2 and 3 at a distance of 100 Mpc, while with the proposed Einstein Telescope, EOS parameters can be extracted to accuracy an order of magnitude better.

astro-ph.HE↗

arXiv2010

Inferring the neutron star equation of state from binary inspiral waveforms

The properties of neutron star matter above nuclear density are not precisely known. Gravitational waves emitted from binary neutron stars during their late stages of inspiral and merger contain imprints of the neutron-star equation of state. Measuring departures from the point-particle limit of the late inspiral waveform allows one to measure properties of the equation of state via gravitational wave observations. This and a companion talk by J. S. Read reports a comparison of numerical waveforms from simulations of inspiraling neutron-star binaries, computed for equations of state with varying stiffness. We calculate the signal strength of the difference between waveforms for various commissioned and proposed interferometric gravitational wave detectors and show that observations at frequencies around 1 kHz will be able to measure a compactness parameter and constrain the possible neutron-star equations of state.

gr-qc↗

arXiv2009

Non-conformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.

gr-qc↗

arXiv2009

Iteration Stability for Simple Newtonian Stellar Systems

For an equation of state in which pressure is a function only of density, the analysis of Newtonian stellar structure is simple in principle if the system is axisymmetric, or consists of a corotating binary. It is then required only to solve two equations: one stating that the "injection energy", $κ$, a potential, is constant throughout the stellar fluid, and the other being the integral over the stellar fluid to give the gravitational potential. An iterative solution of these equations generally diverges if $κ$ is held fixed, but converges with other choices. We investigate the mathematical reason for this convergence/divergence by starting the iteration from an approximation that is perturbatively different from the actual solution. A cycle of iteration is then treated as a linear "updating" operator, and the properties of the linear operator, especially its spectrum, determine the convergence properties. For simplicity, we confine ourselves to spherically symmetric models in which we analyze updating operators both in the finite dimensional space corresponding to a finite difference representation of the problem, and in the continuum, and we find that the fixed-$κ$ operator is self-adjoint and generally has an eigenvalue greater than unity; in the particularly important case of a polytropic equation of state with index greater than unity, we prove that there must be such an eigenvalue. For fixed central density, on the other hand, we find that the updating operator has only a single eigenvector, with zero eigenvalue, and is nilpotent in finite dimension, thereby giving a convergent solution.

astro-ph.SR↗

arXiv2009

Measuring the neutron star equation of state with gravitational wave observations

We report the results of a first study that uses numerical simulations to estimate the accuracy with which one can use gravitational wave observations of double neutron star inspiral to measure parameters of the neutron-star equation of state. The simulations use the evolution and initial-data codes of Shibata and Uryu to compute the last several orbits and the merger of neutron stars, with matter described by a parametrized equation of state. Previous work suggested the use of an effective cutoff frequency to place constraints on the equation of state. We find, however, that greater accuracy is obtained by measuring departures from the point-particle limit of the gravitational waveform produced during the late inspiral. As the stars approach their final plunge and merger, the gravitational wave phase accumulates more rapidly for smaller values of the neutron star compactness (the ratio of the mass of the neutron star to its radius). We estimate that realistic equations of state will lead to gravitational waveforms that are distinguishable from point particle inspirals at an effective distance (the distance to an optimally oriented and located system that would produce an equivalent waveform amplitude) of 100 Mpc or less. As Lattimer and Prakash observed, neutron-star radius is closely tied to the pressure at density not far above nuclear. Our results suggest that broadband gravitational wave observations at frequencies between 500 and 1000 Hz will constrain this pressure, and we estimate the accuracy with which it can be measured. Related first estimates of radius measurability show that the radius can be determined to an accuracy of ~1 km at 100 Mpc.

gr-qc↗